Problem: One more than $11$ times a certain prime $p$ is another prime $q$. What is the value of $q$?
Solution: We try plugging in values for $p$ and see if $11p+1$ is a prime number. The smallest prime number is $2$, so we try $11(2)+1=23$, which is prime. The value of $q$ is $\boxed{23}$.